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Author Archives: jgl
On the word topology, and beyond
Today I (Jean G.L.) have the pleasure to have a guest, Aliaume Lopez. We are going to talk about the word topology on X*. In the book, there is a socalled Topological Higman Lemma that says that, if X is … Continue reading
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Chains and nested spaces
A chain is a totally ordered poset, and a nested space is a topological space whose lattice of open sets is a chain. That may seem like a curious notion, although you might say that the Scott topology on the … Continue reading
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TD spaces
In any topological space, the closure of any oneelement set {x} is also its downward closure ↓x with respect to the specialization preordering. A TD space is a topological space in which, for every point x, ↓x – {x} is … Continue reading
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Zhao, Xi and Chen’s wellfiltered, nonsober dcpo
There are several known examples of dcpos that are wellfiltered, but not sober, and I have already mentioned one due to Xiaodong Jia. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This … Continue reading
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QuasiPolish spaces as rounded ideal completions
This month, a pearl by Matthew de Brecht. It is known that the rounded ideal completion of an abstract basis (a set B with a transitive, interpolative relation) is a continuous dcpo, and that all continuous dcpos can be obtained … Continue reading
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Convergence without points
Can you define convergence without mentioning points? More precisely, is there any form of Stone duality for convergence spaces, instead of just topological spaces? The short answer is yes. For the long answer, read the full post.
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X. Jia’s wellfiltered, nonsober dcpo
[Business as usual, despite all viruses!] Peter Johnstone once showed the existence of a dcpo J that is not sober in its Scott topology. That dcpo is not wellfiltered either. Is there a dcpo that is not sober but is … Continue reading
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Dcpos built as graphs of functions
Let X and P be two dcpos, and let ψ be a map from X to P. When is the graph of ψ a dcpo? I will give you a funny sufficient condition, which involves the socalled dtopology, and Hausdorffness. … Continue reading
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Borel sets, analytic sets, and the Baire property
Let me venture into the realm of σalgebras. Yes, you might say, that is measure theory, not topology… but topology plays an important role in measure theory and, for that matter, descriptive set theory. I will tell you about sets … Continue reading
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The RasiowaSikorski lemma and the Baire property
Merry Christmas! And a Happy New Year, too. There are incredible links between logic and topology, and I would like to start with something called the RasiowaSikorski lemma in logic. That is a theorem that states the existence of certain … Continue reading
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